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5. Working modulo 9 or 11, find the missing digits in the calculations below: (a) 51840 273581 = 1418243x040. (b) 2x99561 = [3(523 + x)]². (c) 2784x = x 5569. %3D %3D

5. Working modulo 9 or 11, find the missing digits in the calculations below: (a) 51840 273581 = 1418243x040. (b) 2x99561 = [3(523 + x)]². (c) 2784x = x 5569. %3D %3D

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 55E: The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for...
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6a,b,c
(b) Give criteria for the divisibility of N by 3 and 8 that depend on the digits of N 4. For any n > 1. prove that there exists a prime with at least n of its digits equal to 0. [Hint: Consider the arithmetic progression 10"+1k +1 for k = 1, 2, ....] M = (100a2+ 10a1 + ao) – (100a5 +10a4+ a3) 6. Show that 2" divides an integer N if and only if 2" divides the number made up of the 5. Find the values of n > 1 for which 1!+ 2! + 3! + ·. .+n! is a perfect square. 7. Let N = am 10m + + a2102 + a¡ 10+ ao, where 0<ak <9, be the decimal expan- (a) Prove that 7, 11, and 13 all divide N if and only if 7, 11, and 13 divide the integer written in the base 9. (c) Is the integer (447836)9 divisible by 3 and 8? 6. Working modulo 9 or 11, find the missing digits in the calculations below: (a) 51840 273581 = 1418243x040. (b) 2x99561 = [3(523 + x)]². (c) 2784x =x 5569. (d) 512 1x53125 = 1000000000. 7. Establish the following divisibility criteria: (a) An integer is divisible by 2 if and only if its units digit is 0, 2, 4, 6, or 8. (b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3. (c) An integer is divisible by 4 if and only if the number formed by its tens and u digits is divisible by 4. [Hint: 10 = 0 (mod 4) for k > 2.] (d) An integer is divisible by 5 if and only if its units digit is 0 or 5. 8. For any integer a, show that a? 9. Find the remainder when 44444444 is divided by 9. [Hint: Observe that 2 =-1 (mod 9).] 10. Prove that no integer whose digits add up to 15 can be a square or a cube. [Hint: For any a, a’ = 0, 1, or 8 (mod 9).] 11. Assuming that 495 divides 273x49y5, obtain the digits x and 12. Determine the last three digits of the number 7999 [Hint: 74n = (1+400)" = 1 +400n (mod 1000).] 3. If t, denotes the nth triangular number, show that tn+2k = tn (mod k); hence t and t - a +7 ends in one of the digits 3, 7, or 9. y. must have the same last digit. [Hint: Problem 2(a).] last n digits of N. [Hint: 10k = 2k5k =0 (mod 2") for k > n.] %3D sion of a positive integer N. + (100as + 10a7 + a6) – . .. [Hint: If n is even, then 10n = 1, 10n+1 = 10, 103n+2 1 103n+1 =100 (m -10. 103n+2 100 odd then 103n =
Transcribed Image Text:(b) Give criteria for the divisibility of N by 3 and 8 that depend on the digits of N 4. For any n > 1. prove that there exists a prime with at least n of its digits equal to 0. [Hint: Consider the arithmetic progression 10"+1k +1 for k = 1, 2, ....] M = (100a2+ 10a1 + ao) – (100a5 +10a4+ a3) 6. Show that 2" divides an integer N if and only if 2" divides the number made up of the 5. Find the values of n > 1 for which 1!+ 2! + 3! + ·. .+n! is a perfect square. 7. Let N = am 10m + + a2102 + a¡ 10+ ao, where 0<ak <9, be the decimal expan- (a) Prove that 7, 11, and 13 all divide N if and only if 7, 11, and 13 divide the integer written in the base 9. (c) Is the integer (447836)9 divisible by 3 and 8? 6. Working modulo 9 or 11, find the missing digits in the calculations below: (a) 51840 273581 = 1418243x040. (b) 2x99561 = [3(523 + x)]². (c) 2784x =x 5569. (d) 512 1x53125 = 1000000000. 7. Establish the following divisibility criteria: (a) An integer is divisible by 2 if and only if its units digit is 0, 2, 4, 6, or 8. (b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3. (c) An integer is divisible by 4 if and only if the number formed by its tens and u digits is divisible by 4. [Hint: 10 = 0 (mod 4) for k > 2.] (d) An integer is divisible by 5 if and only if its units digit is 0 or 5. 8. For any integer a, show that a? 9. Find the remainder when 44444444 is divided by 9. [Hint: Observe that 2 =-1 (mod 9).] 10. Prove that no integer whose digits add up to 15 can be a square or a cube. [Hint: For any a, a’ = 0, 1, or 8 (mod 9).] 11. Assuming that 495 divides 273x49y5, obtain the digits x and 12. Determine the last three digits of the number 7999 [Hint: 74n = (1+400)" = 1 +400n (mod 1000).] 3. If t, denotes the nth triangular number, show that tn+2k = tn (mod k); hence t and t - a +7 ends in one of the digits 3, 7, or 9. y. must have the same last digit. [Hint: Problem 2(a).] last n digits of N. [Hint: 10k = 2k5k =0 (mod 2") for k > n.] %3D sion of a positive integer N. + (100as + 10a7 + a6) – . .. [Hint: If n is even, then 10n = 1, 10n+1 = 10, 103n+2 1 103n+1 =100 (m -10. 103n+2 100 odd then 103n =
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